Question

A metal rod, fixed at both ends, has a length $l$,area of cross section $A$ and youngs modulus $Y$ and coefficient of linear expansion $\alpha$ is heated through $t^{0}C$. The work done in heating the rod is

• A. $YA{l}^2{t}^2$
• B. $\frac{1}{4}YAl{\alpha }^2{t}^2$
• C. $\frac{1}{2}YAl{\alpha }^2{t}^2$
• D. $\frac{1}{2}YAl\alpha t$

Answer 1

The correct option is C

$\frac{1}{2}YAl{\alpha }^2{t}^2$

The work done is nothing but the potential energy stored.

Also, the thermal stress is given by $\frac{T}{A}=Y\alpha t$. We have,

$W=\frac{1}{2}\left(stress\right)\left(strain\right)\left(volume\right)$

$W=\frac{1}{2}\times \frac{Y}{l}\times \frac{x^2}{l}\times \left(Al\right)$ ..... (1)
Since $x=\frac{Tl}{YA}$ and $T=YA\alpha t$, we have on substituting in (1),

$W=\frac{1}{2}YAl{\alpha }^2{t}^2$